The generalized Roper-Suffridge extension operator in Banach spaces (II)

In this paper, we generalize the Roper-Suffridge extension operator from Cn to Banach spaces. It is proved that this operator preserves the biholomorphic ɛ starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ɛ starlike mappings on some domains Ω in Cn, or Hilbert spaces, or Banach spaces from univalent ɛ starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in Cn are extended to Hilbert spaces or Banach spaces.